Momentum: Collisions
You're in a car, driving to the gas station. Ahead of you, there is a large truck traveling southbound. Instantly, a small convertible appears to be traveling toward the truck, northbound. The truck driver, who doesn't see the convertible, swerves to the left, hitting the convertible head-on. After the two cars crash, they remain stuck together, and they both travel a few feet southbound, in the original direction of the truck. This act of sticking together after crashing is what we call collisions. But how far to the two cars travel once they crash? The speed and direction of their travel is determined by laws of momentum. General Facts About Momentum The equation for momentum is p=mv. It is typically measured in kg•m/s. Momentum refers to the motion of an object. An object in motion has momentum; it has a speed, and it has a mass. The mass and velocity are directly proportional to an object's momentum; if the velocity or the mass of an object increases, its mometum increases. Momentum is a vector quantity, which simply means that it has a direction. Velocity is very similar to speed, but with velocity, the speed must point in a specific direction. Due to this, momentum must has a direction. That direction is the same direction as the velocity of the object. Newton's Second Law and Momentum Newton's Second Law of Motion recognizes that an unbalanced force on an object causes a change in that object's momentum. Basically, the equation for Newton's Second Law can be re-written to identify a relationship containing momentum. *Fnet=ma -> this is the orignal form of Newton's Second Law, which states that the total force on an object is equivalent to the mass of the object multiplied by its acceleration. *Fnet=(m∆v)/t -> here, acceleration has been rewritten in terms of velocity, since the acceleration of an object is equivalent to the change in its velocity divided by the time it takes to change velocity. *Fnet= ∆(mv)/t -> using mathematical laws of multiplication, the equation can be rewritten to allow for it to be translated to an equation containing momentum. *t•Fnet=∆(mv) -> by multiplying the total force by the time, it is easy to isolated ∆(mv), which is the change in momentum. *t•Fnet=∆p -> the final equation, which states that the change in momentum is equivalent to the total force on an object multiplied by the time in which the change takes place. So what does this all mean? Now, it is a little more clear as to how momentum came about. Sir Issaac Newton, an extraordinary man of his time, created three key laws in physics, among other things. By exmaining the second law as performed above, we can see the relationship between momentum and another concept in physics-general motion! The Law of Conservation of Momentum Things to Remember These fun little images can be found at www.batesville.k12.in.us ---- ---- ---- ---- General Sample Problems Based on the information above, let's see if you can answer some general questions on momentum: a. A 1000-kg car moving 20 m/s northward b. A 54-kg student moving 2 m/s westward c. A car has 2000 kg•m/s of momentum. What would happen if the velocity were doubled? If the mass were tripled? Solutions Collisions and the Conservation of Momentum In all situations, momentum must be conserved. This is where the Law of the Conservation of Momentum comes in. This law states that the total momentum in the initial situation is exactly equal to that of the final situation. Simply speaking, when two objects collide, the "before" picture must have the same total momentum as the "final" picture does. Elastic Collisions So what exactly are elastic collisions? As we learned before, a collision is simply when to objects hit one another, essentially colliding. An elastic collision, though, relates to the final picutre of a collision. What happens after two cars hit each other? In an elastic collision, the two cars, or two object, sort of "bounce off" of one another. It's almost as if someone placed a spring in between the two objects so that they would just bounce right back off of each other. A perfect example of an elastic collision can be found in a game of pool. When you hit the white ball, it probably hits one of the striped or solid balls, if you have good aim. What happens once the two balls collide? The colored ball and the white ball, if you watch them carefully, do not travel together in the initial direction of the white ball. What tends to happen is that the white ball comes to a halt, following it's inital path, whereas the colored ball continues to travel in the direction of the white ball. By using the Law of the Conservation of Momentum, we can examine a situation like this one in great detail. Given the image below, we can see that ball 1 and ball 2 have the same mass of 0.5 kg. Ball 1 is travelling at 1.0 m/s eastward, and ball 2 is travelling at 2.0m/s westward. As we determined before, the momentum of every object can be determined by multiplying the objects velocity and mass. The momentums of both balls in the collision are as follows: *p=mv **ball 1: p=(0.5 kg)(1.0 m/s East) ***ball 1: p= 0.5 kg•m/s East **ball 2: p=(0.5 kg)(2.0 m/s West) ***ball 2: p=1.0 kg•m/s West Consequently, the cumulative momentum for the entire picture is as follows *p(ball 1) + p(ball 2)= p(total) *0.5 kg•m/s + 1.0 kg•m/s= p(total) *1.5 kg•m/s= p(total) So what happens when they collide in an elastic collision? Let's say that ball 1 bounces off of ball 2, and after the collision, it is travelling WESTWARD at 1.5 m/s. That means that the direction as well as the magnitude of the velocity have changed. But remember: the mass of ball 1 is the same. Ball 2 has the same mass as it had before, but now we want to figure out it's final velocity. So we'll use what we know to figure out what we don't know. We can use the mass and final velocity of ball 1 to determine it's momentum: *p=mv **p=(0.5 kg)(1.5 m/s West) **p=0.75 kg•m/s West According to the Law of Conservation of Momentum, the final momentum of the entire scene is equivalent to the initial momentum of the entire scene. If our inital cumulative momentum is 1.5 kg•m/s, that means that our final momentum has to equal 1.5 kg•m/s as well. *p(initial)=p(final) *1.5 kg•m/s=p(final) We know have to use our prior knowledge to figure out the rest of this problem. We'll do it step by step. We know that the final momentum is the sum of an addition problem: *p(final)=p(ball 1 final) + p(ball 2 final) We know parts of this equation already! Let's fill in what we know: *1.5 kg•m/s= 0.75 kg•m/s + p(ball 2 final) Doing an easy form of algebra, we can conclude that the momentum of ball 2 is 0.75 kg•m/s: *1.5 kg•m/s= 0.75 kg•m/s + p(ball 2 final) *0.75 kg•m/s= p(ball 2 final) We also know that momentum is equivalent to mass times velocity, so we can rewrite this equation with that information: *0.75 kg•m/s= mv Because we know that the mass of ball 2 remains unchanged, we can plug in 0.5 kg: *0.75 kg•m/s= (0.5 kg)v Again, algebra will help us to isolate the velocity, and conclude the magnitude of it: *0.75 kg•m/s= (0.5 kg)v *1.5 m/s=v And there we go! We've found the final speed of ball 2, and in order to make it a velocity, we have to figure out the direction of the ball. Because we know that the direction of the ball reverses after the collision, and this ball was travelling west prior to colliding, we can conclude that the velocity of ball 2 in its final state is 1.5 m/s east. Inelastic Collisions Sample Regents Problems Sample National Exam Problems 'Resources' For more information on momentum and collisions, please feel free to browse the following websites: